{"paper":{"title":"Finite Potential Energy for Entire Solutions of the Planar Ginzburg--Landau Equation","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Haicheng Yan, Hongge Chen, Juncheng Wei, Wen Yang","submitted_at":"2026-07-20T02:32:20Z","abstract_excerpt":"We prove that every smooth entire solution $ u\\colon\\mathbb{R}^2\\to\\mathbb{R}^2 $ of the Ginzburg--Landau equation $ -\\Delta u=u(1-|u|^2) $ with $ |u(x)|\\to1 $ as $ |x|\\to\\infty $ has finite potential energy, i.e., \\begin{equation*} \\int_{\\mathbb{R}^2}(1-|u|^2)^2\\dd x<+\\infty, \\end{equation*} thereby resolving Brezis' Open Problem 2.5 in \\cite{BrezisProblems}. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like $ |x|^{-1} $; such a mode lies outside $ L^2 $ and does not admit a single-valued potential. By minimizing ove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17490","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.17490/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}